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Unformatted text preview: 10Jan2011 Chemistry 21b – Spectroscopy Lecture # 4 – The Born-Oppenheimer Approximation, H + 2 1. Born-Oppenheimer approximation As for atoms, all information about a molecule is contained in the wave function Ψ, which is the solution of the time-independent Schr¨ odinger equation: ˆ H Ψ( −→ x , −→ r ) = E Ψ( −→ x , −→ r ) (4 . 1) where −→ x stands collectively for the spatial and spin coordinates of the n electrons in the molecule, and −→ r denotes collectively the positions of all N nuclei in the molecule. In the non-relativistic limit, the total Hamiltonian for the molecule is ˆ H = ˆ T N + ˆ T e + ˆ V Ne + ˆ V ee + ˆ V NN (4 . 2) ≡ ˆ T N + ˆ H el where ˆ T N = − summationdisplay α (1 / 2) M α ∇ 2 α , ˆ T e = − summationdisplay i (1 / 2) ∇ 2 i (4 . 3 − 4) ˆ V Ne = − summationdisplay α,i Z α | R α − r i | , ˆ V ee = summationdisplay i>j 1 | r i − r j | , ˆ V NN = summationdisplay α>β Z α Z β | R α − R β | (4 . 5 − 7) Atomic units have been used, in which ¯ h = m e = e = 1 . ˆ T N and ˆ T e are the summed kinetic energy operators of the nuclei α with mass M α and the electrons i with mass m e , respectively, and ˆ V Ne , ˆ V ee and ˆ V NN denote the summed Coulomb interaction energies between the nuclei and the electrons, between the electrons themselves, and between the nuclei themselves, respectively. Equation (4.1) is a (3 n +3 N )- dimensional second order partial differential equation, which cannot be readily solved. Because the masses of the nuclei are much larger than that of the electrons, the nuclei move slowly compared with the electrons. It is usually (but not always!) a very good approximation to assume that the electronic energies (that is, the energies due to the motions of the electrons) can be determined accurately with the nuclei held fixed at each possible set of nuclear positions. In other words, it is assumed that the electrons adjust adiabatically to small or slow changes in the nuclear geometry. This approximation and its consequences were first examined by Born and Oppenheimer (1927, Ann. Physik 85 , 457), and has carried their names ever since. In this approximation, the total wave function is separable Ψ( −→ x , −→ R ) = Ψ el ( −→ x ; −→ R )Ψ nuc ( −→ R ) (4 . 8) into a nuclear part Ψ nuc that depends only upon the nuclear coordinates −→ R , and an electronic part Ψ el that depends on the electronic coordinates −→ x , but only parametrically on −→ R . Ψ el is the solution of the electronic eigenvalue equation ˆ H el Ψ el ( −→ x ; −→ R ) = E el ( −→ R )Ψ el ( −→ x ; −→ R ) (4 . 9) 34 where E el ( R ) is the potential energy surface, or, in the case of a diatomic molecule, the potential energy curve of the molecule in a particular electronic state. Substituting the Born-Oppenheimer wavefunction (7.8) into the Schr¨ odinger equation, and using (7.9) gives: bracketleftbigg − summationdisplay...
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